Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

##       vmr             pmr             mmr             vhr       
##  Min.   :0.868   Min.   :0.904   Min.   :0.988   Min.   :0.849  
##  1st Qu.:1.044   1st Qu.:1.042   1st Qu.:1.013   1st Qu.:1.039  
##  Median :1.097   Median :1.084   Median :1.085   Median :1.099  
##  Mean   :1.070   Mean   :1.065   Mean   :1.066   Mean   :1.085  
##  3rd Qu.:1.136   3rd Qu.:1.107   3rd Qu.:1.101   3rd Qu.:1.160  
##  Max.   :1.168   Max.   :1.141   Max.   :1.133   Max.   :1.214  
##       phr             mhr       
##  Min.   :0.878   Min.   :0.977  
##  1st Qu.:1.068   1st Qu.:1.013  
##  Median :1.128   Median :1.113  
##  Mean   :1.095   Mean   :1.087  
##  3rd Qu.:1.182   3rd Qu.:1.128  
##  Max.   :1.208   Max.   :1.207
##       vmrl      
##  Min.   :0.801  
##  1st Qu.:1.013  
##  Median :1.085  
##  Mean   :1.061  
##  3rd Qu.:1.128  
##  Max.   :1.193
##            vmr   pmr   mmr   vhr   phr   mhr
## Min.   : 0.868 0.904 0.988 0.849 0.878 0.977
## 1st Qu.: 1.044 1.042 1.013 1.039 1.068 1.013
## Median : 1.097 1.084 1.085 1.099 1.128 1.113
## Mean   : 1.070 1.065 1.066 1.085 1.095 1.087
## 3rd Qu.: 1.136 1.107 1.101 1.160 1.182 1.128
## Max.   : 1.168 1.141 1.133 1.214 1.208 1.207

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.095 phr 1.136 vmr 1.168 vmr
0.977 mhr 1.044 vmr 1.113 mhr 1.087 mhr 1.107 pmr 1.141 pmr
0.904 pmr 1.042 pmr 1.099 vhr 1.085 vhr 1.101 mmr 1.133 mmr
0.878 phr 1.039 vhr 1.097 vmr 1.070 vmr 1.160 vhr 1.214 vhr
0.868 vmr 1.013 mmr 1.085 mmr 1.066 mmr 1.182 phr 1.208 phr
0.849 vhr 1.013 mhr 1.084 pmr 1.065 pmr 1.128 mhr 1.207 mhr

Covariance

## cov(vmr, pmr) =  -0.001094875
## cov(vhr, phr) =  -0.0001730651

Velliv medium risk, 2011 - 2023

## 
## AIC: -27.8497 
## BIC: -25.58991 
## m: 0.0480931 
## s: 0.1198426 
## nu (df): 3.303595 
## xi: 0.03361192 
## R^2: 0.993 
## 
## An R^2 of 0.993 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 7.4 percent
## What is the risk of losing max 25 %? =< 1.8 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 41 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 279.25 kr.
## SD of portfolio index value after 20 years: 122.605 kr.
## Min total portfolio index value after 20 years: 0.794 kr.
## Max total portfolio index value after 20 years: 869.201 kr.
## 
## Share of paths finishing below 100: 4.88 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Velliv medium risk, 2007 - 2023

Fit to skew t distribution

## 
## AIC: -34.35752 
## BIC: -31.02467 
## m: 0.05171176 
## s: 0.1149408 
## nu (df): 2.706099 
## xi: 0.5049945 
## R^2: 0.978 
## 
## An R^2 of 0.978 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.4 percent
## What is the risk of losing max 25 %? =< 1.3 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 36.2 percent
## What is the chance of gaining min 25 %? >= 0.3 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 295.768 kr.
## SD of portfolio index value after 20 years: 118.405 kr.
## Min total portfolio index value after 20 years: 0.643 kr.
## Max total portfolio index value after 20 years: 996.677 kr.
## 
## Share of paths finishing below 100: 3.16 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Velliv high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -21.42488 
## BIC: -19.16508 
## m: 0.06471454 
## s: 0.1499924 
## nu (df): 3.144355 
## xi: 0.002367034 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 8.3 percent
## What is the risk of losing max 25 %? =< 2.5 percent
## What is the risk of losing max 50 %? =< 0.4 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 53.3 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.02 percent
## 
## Mean portfolio index value after 20 years: 404.935 kr.
## SD of portfolio index value after 20 years: 218.273 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1475.734 kr.
## 
## Share of paths finishing below 100: 4.25 percent

Max vs sum plots

Max vs sum plots for the first four moments:

PFA medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -33.22998 
## BIC: -30.97018 
## m: 0.05789224 
## s: 0.1234592 
## nu (df): 2.265273 
## xi: 0.477324 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.9 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 32.7 percent
## What is the chance of gaining min 25 %? >= 0.1 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.

##  [1] -0.091256521 -0.003731241  0.027312079  0.045808232  0.059068633
##  [6]  0.069575113  0.078454727  0.086316936  0.093536451  0.100370932
## [11]  0.107018607  0.114081432  0.127604387

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 321.547 kr.
## SD of portfolio index value after 20 years: 107.596 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1763.269 kr.
## 
## Share of paths finishing below 100: 2.16 percent

Max vs sum plots

Max vs sum plots for the first four moments:

PFA high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -23.72565 
## BIC: -21.46585 
## m: 0.08386034 
## s: 0.1210107 
## nu (df): 3.184569 
## xi: 0.01790306 
## R^2: 0.964 
## 
## An R^2 of 0.964 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.3 percent
## What is the risk of losing max 25 %? =< 1.4 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 59.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.02 percent
## 
## Mean portfolio index value after 20 years: 550.001 kr.
## SD of portfolio index value after 20 years: 242.77 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1901.112 kr.
## 
## Share of paths finishing below 100: 0.95 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Mix medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -36.9603 
## BIC: -34.7005 
## m: 0.05902873 
## s: 0.08757749 
## nu (df): 2.772621 
## xi: 0.02904471 
## R^2: 0.89 
## 
## An R^2 of 0.89 suggests that the fit is not completely random.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.7 percent
## What is the risk of losing max 50 %? =< 0.1 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 35.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The fit suggests big losses for the lowest percentiles, which are not present in the data.
So the fit is actually a very cautious estimate.

Data vs fit

Let’s plot the fit and the observed returns together.

Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 323.587 kr.
## SD of portfolio index value after 20 years: 98.626 kr.
## Min total portfolio index value after 20 years: 0.431 kr.
## Max total portfolio index value after 20 years: 669.953 kr.
## 
## Share of paths finishing below 100: 1.23 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 301.929 kr.
## SD of portfolio index value after 20 years: 81.325 kr.
## Min total portfolio index value after 20 years: 4.07 kr.
## Max total portfolio index value after 20 years: 634.451 kr.
## 
## Share of paths finishing below 100: 0.33 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Mix high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -24.26084 
## BIC: -22.00104 
## m: 0.0822419 
## s: 0.07129843 
## nu (df): 89.86289 
## xi: 0.7697502 
## R^2: 0.961 
## 
## An R^2 of 0.961 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0.9 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 46.1 percent
## What is the chance of gaining min 25 %? >= 1.2 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that the high risk mix provides a much better upside and smaller downside.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 500.501 kr.
## SD of portfolio index value after 20 years: 156.921 kr.
## Min total portfolio index value after 20 years: 136.926 kr.
## Max total portfolio index value after 20 years: 1342.051 kr.
## 
## Share of paths finishing below 100: 0 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 477.534 kr.
## SD of portfolio index value after 20 years: 161.989 kr.
## Min total portfolio index value after 20 years: 71.406 kr.
## Max total portfolio index value after 20 years: 1216.783 kr.
## 
## Share of paths finishing below 100: 0.06 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Compare pension plans

Risk of max loss

Risk of max loss of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 21.3 18.2 19.9 12.2 14.3 12.7 13.0
5 12.5 9.6 12.8 6.0 8.6 6.2 4.2
10 7.4 5.4 8.3 3.3 5.3 3.3 0.9
25 1.8 1.3 2.5 0.9 1.4 0.7 0.0
50 0.2 0.2 0.4 0.2 0.2 0.1 0.0
90 0.0 0.0 0.0 0.0 0.0 0.0 0.0
99 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Worst ranking for loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.3 Velliv_m 12.8 Velliv_h 8.3 Velliv_h 2.5 Velliv_h 0.4 Velliv_h 0 Velliv_m 0 Velliv_m
19.9 Velliv_h 12.5 Velliv_m 7.4 Velliv_m 1.8 Velliv_m 0.2 Velliv_m 0 Velliv_m_l 0 Velliv_m_l
18.2 Velliv_m_l 9.6 Velliv_m_l 5.4 Velliv_m_l 1.4 PFA_h 0.2 Velliv_m_l 0 Velliv_h 0 Velliv_h
14.3 PFA_h 8.6 PFA_h 5.3 PFA_h 1.3 Velliv_m_l 0.2 PFA_m 0 PFA_m 0 PFA_m
13.0 mix_h 6.2 mix_m 3.3 PFA_m 0.9 PFA_m 0.2 PFA_h 0 PFA_h 0 PFA_h
12.7 mix_m 6.0 PFA_m 3.3 mix_m 0.7 mix_m 0.1 mix_m 0 mix_m 0 mix_m
12.2 PFA_m 4.2 mix_h 0.9 mix_h 0.0 mix_h 0.0 mix_h 0 mix_h 0 mix_h

Chance of min gains

Chance of min gains of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 78.7 81.8 80.1 87.8 85.7 87.3 87.0
5 63.8 64.9 69.2 71.5 75.8 71.4 69.9
10 41.0 36.2 53.3 32.7 59.6 35.6 46.1
25 0.0 0.3 0.0 0.1 0.0 0.0 1.2
50 0.0 0.0 0.0 0.0 0.0 0.0 0.0
100 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Best ranking for gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
87.8 PFA_m 75.8 PFA_h 59.6 PFA_h 1.2 mix_h 0 Velliv_m 0 Velliv_m
87.3 mix_m 71.5 PFA_m 53.3 Velliv_h 0.3 Velliv_m_l 0 Velliv_m_l 0 Velliv_m_l
87.0 mix_h 71.4 mix_m 46.1 mix_h 0.1 PFA_m 0 Velliv_h 0 Velliv_h
85.7 PFA_h 69.9 mix_h 41.0 Velliv_m 0.0 Velliv_m 0 PFA_m 0 PFA_m
81.8 Velliv_m_l 69.2 Velliv_h 36.2 Velliv_m_l 0.0 Velliv_h 0 PFA_h 0 PFA_h
80.1 Velliv_h 64.9 Velliv_m_l 35.6 mix_m 0.0 PFA_h 0 mix_m 0 mix_m
78.7 Velliv_m 63.8 Velliv_m 32.7 PFA_m 0.0 mix_m 0 mix_h 0 mix_h

MC risk percentiles

Risk of loss from first to last period.

_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

_m is medium.
_h is high.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 4.88 3.16 4.25 2.16 0.95 1.23 0 0.33 0.06
5 4.28 2.82 3.78 1.94 0.86 1.12 0 0.25 0.04
10 3.71 2.50 3.28 1.76 0.78 1.00 0 0.17 0.03
25 2.26 1.70 2.34 1.26 0.57 0.69 0 0.09 0.01
50 0.84 0.85 1.26 0.71 0.25 0.34 0 0.02 0.00
90 0.03 0.10 0.17 0.19 0.07 0.02 0 0.01 0.00
99 0.01 0.01 0.04 0.04 0.03 0.01 0 0.00 0.00

Worst ranking for MC loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.88 Velliv_m 4.28 Velliv_m 3.71 Velliv_m 2.34 Velliv_h 1.26 Velliv_h 0.19 PFA_m 0.04 Velliv_h
4.25 Velliv_h 3.78 Velliv_h 3.28 Velliv_h 2.26 Velliv_m 0.85 Velliv_m_l 0.17 Velliv_h 0.04 PFA_m
3.16 Velliv_m_l 2.82 Velliv_m_l 2.50 Velliv_m_l 1.70 Velliv_m_l 0.84 Velliv_m 0.10 Velliv_m_l 0.03 PFA_h
2.16 PFA_m 1.94 PFA_m 1.76 PFA_m 1.26 PFA_m 0.71 PFA_m 0.07 PFA_h 0.01 Velliv_m
1.23 mix_m_a 1.12 mix_m_a 1.00 mix_m_a 0.69 mix_m_a 0.34 mix_m_a 0.03 Velliv_m 0.01 Velliv_m_l
0.95 PFA_h 0.86 PFA_h 0.78 PFA_h 0.57 PFA_h 0.25 PFA_h 0.02 mix_m_a 0.01 mix_m_a
0.33 mix_m_b 0.25 mix_m_b 0.17 mix_m_b 0.09 mix_m_b 0.02 mix_m_b 0.01 mix_m_b 0.00 mix_h_a
0.06 mix_h_b 0.04 mix_h_b 0.03 mix_h_b 0.01 mix_h_b 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_m_b
0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_b 0.00 mix_h_b 0.00 mix_h_b

MC gains percentiles

Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 95.12 96.84 95.75 97.84 99.05 98.77 100.00 99.67 99.94
5 94.48 96.44 95.31 97.58 98.93 98.61 100.00 99.55 99.91
10 93.76 95.96 94.82 97.32 98.85 98.52 100.00 99.50 99.87
25 91.09 94.35 93.27 96.54 98.49 97.96 100.00 99.05 99.77
50 85.72 90.37 90.58 94.53 97.49 96.16 99.98 97.72 99.42
100 71.47 78.49 83.31 87.31 94.93 89.45 99.63 90.07 97.65
200 39.82 45.78 63.84 58.49 85.03 59.86 93.02 49.39 87.15
300 16.13 18.28 44.93 22.89 71.02 21.77 71.73 11.53 65.57
400 5.24 5.17 29.35 4.00 54.13 3.62 44.55 1.28 41.34
500 1.04 1.10 17.61 0.50 38.21 0.20 22.94 0.09 21.29
1000 0.00 0.00 0.60 0.02 2.44 0.00 0.26 0.00 0.06

Best ranking for MC gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 99.98 mix_h_a 99.63 mix_h_a
99.94 mix_h_b 99.91 mix_h_b 99.87 mix_h_b 99.77 mix_h_b 99.42 mix_h_b 97.65 mix_h_b
99.67 mix_m_b 99.55 mix_m_b 99.50 mix_m_b 99.05 mix_m_b 97.72 mix_m_b 94.93 PFA_h
99.05 PFA_h 98.93 PFA_h 98.85 PFA_h 98.49 PFA_h 97.49 PFA_h 90.07 mix_m_b
98.77 mix_m_a 98.61 mix_m_a 98.52 mix_m_a 97.96 mix_m_a 96.16 mix_m_a 89.45 mix_m_a
97.84 PFA_m 97.58 PFA_m 97.32 PFA_m 96.54 PFA_m 94.53 PFA_m 87.31 PFA_m
96.84 Velliv_m_l 96.44 Velliv_m_l 95.96 Velliv_m_l 94.35 Velliv_m_l 90.58 Velliv_h 83.31 Velliv_h
95.75 Velliv_h 95.31 Velliv_h 94.82 Velliv_h 93.27 Velliv_h 90.37 Velliv_m_l 78.49 Velliv_m_l
95.12 Velliv_m 94.48 Velliv_m 93.76 Velliv_m 91.09 Velliv_m 85.72 Velliv_m 71.47 Velliv_m
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
93.02 mix_h_a 71.73 mix_h_a 54.13 PFA_h 38.21 PFA_h 2.44 PFA_h
87.15 mix_h_b 71.02 PFA_h 44.55 mix_h_a 22.94 mix_h_a 0.60 Velliv_h
85.03 PFA_h 65.57 mix_h_b 41.34 mix_h_b 21.29 mix_h_b 0.26 mix_h_a
63.84 Velliv_h 44.93 Velliv_h 29.35 Velliv_h 17.61 Velliv_h 0.06 mix_h_b
59.86 mix_m_a 22.89 PFA_m 5.24 Velliv_m 1.10 Velliv_m_l 0.02 PFA_m
58.49 PFA_m 21.77 mix_m_a 5.17 Velliv_m_l 1.04 Velliv_m 0.00 Velliv_m
49.39 mix_m_b 18.28 Velliv_m_l 4.00 PFA_m 0.50 PFA_m 0.00 Velliv_m_l
45.78 Velliv_m_l 16.13 Velliv_m 3.62 mix_m_a 0.20 mix_m_a 0.00 mix_m_a
39.82 Velliv_m 11.53 mix_m_b 1.28 mix_m_b 0.09 mix_m_b 0.00 mix_m_b

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
m 0.048 0.052 0.065 0.058 0.084 0.059 0.082
s 0.120 0.115 0.150 0.123 0.121 0.088 0.071
nu 3.304 2.706 3.144 2.265 3.185 2.773 89.863
xi 0.034 0.505 0.002 0.477 0.018 0.029 0.770
R-squared 0.993 0.978 0.991 0.991 0.964 0.890 0.961

Fit statistics ranking

m ranking s ranking R-squared ranking
0.084 PFA_high 0.071 mix_high 0.993 Velliv_medium
0.082 mix_high 0.088 mix_medium 0.991 Velliv_high
0.065 Velliv_high 0.115 Velliv_medium_long 0.991 PFA_medium
0.059 mix_medium 0.120 Velliv_medium 0.978 Velliv_medium_long
0.058 PFA_medium 0.121 PFA_high 0.964 PFA_high
0.052 Velliv_medium_long 0.123 PFA_medium 0.961 mix_high
0.048 Velliv_medium 0.150 Velliv_high 0.890 mix_medium

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
mc_m 279.250 295.768 404.935 321.547 550.001 323.587 301.929 500.501 477.534
mc_s 122.605 118.405 218.273 107.596 242.770 98.626 81.325 156.921 161.989
mc_min 0.794 0.643 0.000 0.000 0.000 0.431 4.070 136.926 71.406
mc_max 869.201 996.677 1475.734 1763.269 1901.112 669.953 634.451 1342.051 1216.783
dao_pct 0.000 0.000 0.020 0.010 0.020 0.000 0.000 0.000 0.000
losing_pct 4.880 3.160 4.250 2.160 0.950 1.230 0.330 0.000 0.060

Ranking

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking losing_pct ranking
550.001 PFA_h 81.325 mix_m_b 136.926 mix_h_a 1901.112 PFA_h 0.00 Velliv_m 0.00 mix_h_a
500.501 mix_h_a 98.626 mix_m_a 71.406 mix_h_b 1763.269 PFA_m 0.00 Velliv_m_l 0.06 mix_h_b
477.534 mix_h_b 107.596 PFA_m 4.070 mix_m_b 1475.734 Velliv_h 0.00 mix_m_a 0.33 mix_m_b
404.935 Velliv_h 118.405 Velliv_m_l 0.794 Velliv_m 1342.051 mix_h_a 0.00 mix_m_b 0.95 PFA_h
323.587 mix_m_a 122.605 Velliv_m 0.643 Velliv_m_l 1216.783 mix_h_b 0.00 mix_h_a 1.23 mix_m_a
321.547 PFA_m 156.921 mix_h_a 0.431 mix_m_a 996.677 Velliv_m_l 0.00 mix_h_b 2.16 PFA_m
301.929 mix_m_b 161.989 mix_h_b 0.000 Velliv_h 869.201 Velliv_m 0.01 PFA_m 3.16 Velliv_m_l
295.768 Velliv_m_l 218.273 Velliv_h 0.000 PFA_m 669.953 mix_m_a 0.02 Velliv_h 4.25 Velliv_h
279.250 Velliv_m 242.770 PFA_h 0.000 PFA_h 634.451 mix_m_b 0.02 PFA_h 4.88 Velliv_m

Comments

(Ignoring mhr_a…)

mhr has some nice properties:
- It has a relatively high \(nu\) value of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds have \(nu\) values close to 3, except phr which is even worse at close to 2. (Note that for a Gaussian, \(nu\) is infinite.)
- It has the lowest losing percentage of all simulations, which is better than 1/6 that of phr.
- It has a DAO percentage of 0, which is the same as mmr, and less than phr.
- Only phr has a higher mc_m.
- It has a smaller mc_s than the individual components, vhr and phr.
- It has the highest xi of all fits, suggesting right skewness. Density plots for vmr, phr and mmr have an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely low xi values. The density plot for mhr is by far the most symmetrical of all the fits.
- Only mmr has as higher mc_min. However, that of mmr is 18 times higher with 62, so mmr is a clear winner here.
- Naturally, it has a mc_max smaller than the individual components, vhr and phr, but ca. 1.5 times higher then mmr.

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent < 4 will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent alpha = 3. Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.005794859 
## s(data_x): 0.3889062 
## m(data_y): 10.49798 
## s(data_y): 3.278807 
## 
## m(data_x + data_y): 5.251885 
## s(data_x + data_y): 1.568944

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
105.220 104.671 7.265 6.794
104.967 105.081 7.633 7.135
105.124 104.704 7.337 6.934
105.142 104.755 7.325 6.945
104.946 104.770 7.293 6.973
104.993 104.919 7.425 7.074
105.148 104.941 7.231 7.234
104.554 105.357 7.544 6.865
104.835 105.071 7.489 6.883
105.004 104.839 7.512 6.750 
##       m_a             m_b             s_a             s_b       
##  Min.   :104.6   Min.   :104.7   Min.   :7.231   Min.   :6.750  
##  1st Qu.:105.0   1st Qu.:104.8   1st Qu.:7.301   1st Qu.:6.870  
##  Median :105.0   Median :104.9   Median :7.381   Median :6.939  
##  Mean   :105.0   Mean   :104.9   Mean   :7.405   Mean   :6.959  
##  3rd Qu.:105.1   3rd Qu.:105.0   3rd Qu.:7.506   3rd Qu.:7.049  
##  Max.   :105.2   Max.   :105.4   Max.   :7.633   Max.   :7.234

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05982   Min.   :0.04716  
##  1st Qu.:0.06565   1st Qu.:0.05903  
##  Median :0.06803   Median :0.06559  
##  Mean   :0.06985   Mean   :0.06663  
##  3rd Qu.:0.07241   3rd Qu.:0.07253  
##  Max.   :0.08367   Max.   :0.09304

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192